Practice Questions

Q66.Let f : (−1, ∞) →R be defined by f(0) = 1 and f(x) = x1 loge(1 + x), x ≠0 . Then the function f (1) Decreases in (−1, 0) and increases in (0, ∞) (2) Increases in (−1, ∞) (3) Increases in (−1, 0) and decreases in (0, ∞) (4) Decreases in (−1, ∞)

Q66.The area (in sq. units) of the region {(x, y) : 0 ≤y ≤x2 + 1, 0 ≤y ≤x + 1, 21 ≤x ≤2} is (1) 23 (2) 79 16 24 (3) 79 (4) 23 16 6

Q66.The area of the region (in sq. units), enclosed by the circle x2 + y2 = 2 which is not common to the region bounded by the parabola y2 = x and the straight line y = x , is (1) 1 6 (24π −1) (2) 13 (6π −1) (3) 1 3 (12π −1) (4) 16 (12π −1) JEE Main 2020 (07 Jan Shift 1) JEE Main Previous Year Paper = ex such that y(0) = 0, then y(1) is

Q66.The integral ∫21 ex. xx (2 + loge x) dx equals : (1) e(4e + 1) (2) 4e2 −1 (3) e(4e −1) (4) e(2e −1)

Q66.If the value of the integral ∫ 01 3 dx is k6 , then k is equal to: (1−x2) 2 JEE Main 2020 (03 Sep Shift 2) JEE Main Previous Year Paper (1) 2√3 + π (2) 2√3 −π (3) 3√2 + π (4) 3√2 −π

Q66.If θ1 and θ2 be respectively the smallest and the largest values of θ in (0, 2π) −{π} which satisfy the equation, θ2 2cot2θ − sin5 θ + 4 = 0 , then ∫ cos23θdθ is equal to: θ1 (1) π (2) 2π 3 3 (3) π 3 + 61 (4) π9

Q66.The area (in sq. units) of the region {(x, y) ∈R2 : x2 ≤y ≤3 −2x}, is. (1) 32 (2) 34 3 3 (3) 29 (4) 31 3 3 JEE Main 2020 (08 Jan Shift 2) JEE Main Previous Year Paper

Q66.Which of the following points lies on the tangent to the curve x4ey + 2√y + 1 = 3 at the point (1, 0)? JEE Main 2020 (05 Sep Shift 2) JEE Main Previous Year Paper (1) (2, 2) (2) (2, 6) (3) (–2, 6) (4) (−2, 4) + C, where C is a constant of integration, then B(θ)A can be:

Q66.If for all real triplets (a, b, c), f(x) = a + bx + cx2; then ∫1 f(x)dx is equal to: 0 JEE Main 2020 (09 Jan Shift 1) JEE Main Previous Year Paper (1) 2{3f(1) + 2f( 12 )} (2) 12 {f(1) + 3f( 12 )} (3) 1 3 {f(0) + f( 12 )} (4) 16 {f(0) + f(1) + 4f( 12 )} dx is equal to:

Q66.The integral ∫( x sin x+cosx x ) 2dx (1) tan x − x sinx x+cossec x x + C (2) sec x + x sinx tanx+cosx x + C (3) sec x − x sinx tanx+cosx x + C (4) tan x + x sinx x+cossec x x + C

Q66.If ∫ cos2 θ(tandθ2θ+sec 2θ) = λ tan θ + 2 loge|f(θ)| + C where C is a constant of integration, then the ordered pair (λ, f(θ)) is equal to: (1) (1, 1 −tan θ) (2) (−1, 1 −tan θ) (3) (−1, 1 + tan θ) (4) (1, 1 + tan θ) JEE Main 2020 (09 Jan Shift 2) JEE Main Previous Year Paper

Q66.If ∫(e2x + 2ex −e−x −1)e(ex+e−x)dx = g(x)e(ex+e−x) (1) e (2) e2 (3) 1 (4) 2 1 2 x dx is :

Q66.The area (in sq. units) of the largest rectangle ABCD whose vertices A and B lie on the x-axis and vertices C and D lie on the parabola, y = x2 −1 below the x-axis, is : (1) 2 (2) 1 3√3 3√3 (3) 4 (4) 4 3 3√3 π

Q66.If ∫ 1 2 = f(x)(1 + sin6 x) λ + c, where c is a constant of integration, then λf( π3 ) is equal to sin3 x(1+sin6 x) 3 JEE Main 2020 (08 Jan Shift 1) JEE Main Previous Year Paper (1) −98 (2) 2 (3) 9 (4) −2 8

Q67.The solution curve of the differential equation, (1 + e−x)(1 + y2) dxdy = y2 which passes through the point (0, 1), is + 2 ) + 2) (1) y2 + 1 = y(loge( 1+e−x2 ) 2) (2) y2 + 1 = y(loge( 1+ex (3) y2 = 1 + y loge( 1+ex2 ) (4) y2 = 1 + y loge( 1+e−x2 )

Q67.If x3dy + xy ⋅dx = x2dy + 2ydx; y(2) = e and x > 1, then y(4) is equal to : (1) √e (2) 1 + √e 2 2 (3) 3 2 √e (4) 23 + √e

Q67.The integral ∫ π3 tan3 x ⋅sin2 3x(2 sec2 x ⋅sin2 3x + 3 tan x ⋅sin 6x)dx is equal to: 6 (1) 18 7 (2) −19 (3) −118 (4) 29 dy y+3x

Q67.Consider a region R = {(x, y) ∈R2 : x2 ≤y ≤2x}. If a line y = α divides the area of region R into two equal parts, then which of the following is true ? (1) α3 −6α2 + 16 = 0 (2) 3α2 −8α3/2 + 8 = 0 (3) 3α2 −8α + 8 = 0 (4) α3 −6α3/2 −16 = 0

Q67.Area (in sq. units) of the region outside |x|2 + |y|3 = 1 and inside the ellipse x24 + y29 = 1 is (1) 6(π −2) (2) 3(π −2) (3) 3(4 −π) (4) 6(4 −π) x, y > 0, y(0) = 1. If y(π) = a

Q67.The value of ∫ −ππ 2 1+esin (1) π4 (2) π (3) π2 (4) 3π2

Q67.The area (in sq. units) of the region A = {(x, y) : x + y ≤1, 2y2 ≥x } (1) 1 (2) 7 3 6 (3) 1 (4) 5 6 6

Q67. x, 0 ≤x < 12 ⎧ Given: f(x) = 2 1 , x = 12 ⎨ 1 ⎩1 −x, 2 < x ≤1 x ∈R. Then, the area (in sq. units) of the region bounded by the curves, y = f(x) and and g(x) = (x −12 ) 2, y = g(x) between the lines 2x = 1 and 2x = √3, is: (1) 3 1 + √34 (2) √34 −13 (3) 2 1 −√34 (4) 21 + √34

Q67.The value of ∫2π0 sin8xx+cos8sin8 x x (1) 2π (2) 2π2 (3) π2 (4) 4π

Q67.If ∫ 5+7 sincosθ−2θ cos2 θ dθ =Aloge B(θ) (1) 2 sin θ+1 (2) 2 sin θ+1 sin θ+3 5(sin θ+3) (3) 5(sin θ+3) (4) 5(2 sin θ+1) 2 sin θ+1 sin θ+3

Q67.The differential equation of the family of curves, x2 = 4b(y + b), b ∈R, is. (1) x(y') 2 = x + 2yy' (2) x(y') 2 = 2yy' −x (3) xy'' = y' (4) x(y')2 = x −2yy' → → →